Problem P. 4735. (April 2015)

P. 4735. The Examining Institute for Cosmic Accidents (EXINCA) sent the following short report to one of its experts: ``One of the discovery spaceships of the Titanium-Devouring Little Green Men has found a perfectly spherical uniform Titanium asteroid. In order to make preparations for the mining, a straight tunnel was built and rails were put into it. The length of the tunnel was the same as the radius of the asteroid, and both ends of the tunnel were on the surface of the planet. Unfortunately one of the mine-carts, although its brake was applied, slipped into the shaft at one end of the tunnel. Initially it speeded up, then it slowed down gradually, turned back and then it stopped exactly at the middle of the tunnel. When it turned back it nearly hit the leader of the miners.'' EXINCA asked the expert to calculate the following:

\(\displaystyle a)\) Where (at what fraction of the length of the tunnel) did the leader of the miners stand?

\(\displaystyle b)\) What could the coefficient of kinetic friction between the wheels of the mine-cart and the rails be?

\(\displaystyle c)\) How long did the mine-cart move if the period of the spacecraft revolving next to the surface of the asteroid is 24 hours? (The asteroid has no atmosphere.)